Since \(\mathbb {E}[^i\mid \mathcal {F}_{i-1}] = {0}\) , \(\mathcal {I}_{1,t}\)  is a martingale.
We hence aim to apply the strong law of large number [1]}, the central limit theorem [1]}, and the Berry-Esseen bound [3]} for martingales to show each result.
We first characterize the conditional covariance of \(\mathcal {I}_{1,t}\)  defined as (cf. [1]})
\(\langle \mathcal {I}_{1}\rangle _t U\sum _{i=0}^{t}\prod _{j=i+1}^{t}{I - \varphi _j\Sigma }\varphi _i^2 U^T\mathbb {E}[^i(^i)^T\mid \mathcal {F}_{i-1}]U\big (\prod _{j=i+1}^{t}{I - \varphi _j\Sigma }\big )^TU^T.\) 
